3.530 \(\int x^2 (a+b x^3)^{2/3} \, dx\)

Optimal. Leaf size=18 \[ \frac {\left (a+b x^3\right )^{5/3}}{5 b} \]

[Out]

1/5*(b*x^3+a)^(5/3)/b

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ \frac {\left (a+b x^3\right )^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^(2/3),x]

[Out]

(a + b*x^3)^(5/3)/(5*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^3\right )^{2/3} \, dx &=\frac {\left (a+b x^3\right )^{5/3}}{5 b}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (a+b x^3\right )^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^(2/3),x]

[Out]

(a + b*x^3)^(5/3)/(5*b)

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fricas [A]  time = 0.54, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

1/5*(b*x^3 + a)^(5/3)/b

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giac [A]  time = 0.17, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

1/5*(b*x^3 + a)^(5/3)/b

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maple [A]  time = 0.01, size = 15, normalized size = 0.83 \[ \frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^(2/3),x)

[Out]

1/5*(b*x^3+a)^(5/3)/b

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maxima [A]  time = 1.33, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

1/5*(b*x^3 + a)^(5/3)/b

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mupad [B]  time = 1.05, size = 14, normalized size = 0.78 \[ \frac {{\left (b\,x^3+a\right )}^{5/3}}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^(2/3),x)

[Out]

(a + b*x^3)^(5/3)/(5*b)

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sympy [A]  time = 0.93, size = 39, normalized size = 2.17 \[ \begin {cases} \frac {a \left (a + b x^{3}\right )^{\frac {2}{3}}}{5 b} + \frac {x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{5} & \text {for}\: b \neq 0 \\\frac {a^{\frac {2}{3}} x^{3}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+a)**(2/3),x)

[Out]

Piecewise((a*(a + b*x**3)**(2/3)/(5*b) + x**3*(a + b*x**3)**(2/3)/5, Ne(b, 0)), (a**(2/3)*x**3/3, True))

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